Ramsey cardinal

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Ramsey cardinals were introduced by Erdős and Hajnal in [1]. A cardinal $\kappa$ is Ramsey if it has the partition property $\kappa\rightarrow (\kappa)^{\lt\omega}_2$. A partition property $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. The more general partition property $\kappa\to(\lambda)^{\lt\omega}_\gamma$ asserts that for every function $F:[\kappa]^{\lt\omega}\to\gamma$ there is $H\subseteq\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant for every $n$, although the value of $F$ on $[H]^n$ may be different for different $n$. Indeed, if $\kappa$ is Ramsey, then $\kappa\rightarrow (\kappa)^{\lt\omega}_\lambda$ for every $\lambda<\kappa$. Ramsey cardinals were named in honor of Frank Ramsey, whose Ramsey theorem for partition properties of $\omega$ motivated the generalizations of these to uncountable cardinals. A Ramsey cardinal $\kappa$ is exactly the $\kappa$-Erdős cardinal.

Ramsey cardinals have a number of other characterizations. They may be characterized model theoretically through the existence of $\kappa$-sized sets of indiscernibles for models meeting the criteria discussed below, as well as through the existence of $\kappa$-sized models of set theory without power set with iterable ultrapowers.

Indiscernibles: Suppose $\mathcal A=(A,\ldots)$ is a model of a language $\mathcal L$ of size less than $\kappa$ whose universe $A$ contains $\kappa$ as a subset.

A cardinal $\kappa$ is Ramsey if and only if every such model $\mathcal A$ has a $\kappa$-sized set of indiscernibles $H\subseteq\kappa$, that is, for every formula $\varphi(\overline x)$ of $\mathcal L$ and every pair of tuples $\overline \alpha$ and $\overline \beta$ of elements of $H$, we have $\mathcal A\models\varphi (\overline \alpha)\leftrightarrow \varphi(\overline \beta)$. [2]

Good sets of indiscernibles: Suppose $A\subseteq\kappa$ and $L_\kappa[A]$ denotes the $\kappa^{\text{th}}$-level of the universe constructible using a predicate for $A$. A set $I\subseteq\kappa$ is a good set of indiscernibles for the model $\langle L_\kappa[A],A\rangle$ if for all $\gamma\in I$,

  • $\langle L_\gamma[A\cap \gamma],A\cap \gamma\rangle\prec \langle L_\kappa[A], A\rangle$,
  • $I\setminus\gamma$ is a set of indiscernibles for the model $\langle L_\kappa[A], A,\xi\rangle_{\xi\in\gamma}$.

A cardinal $\kappa$ is Ramsey if and only if for every $A\subseteq\kappa$, there is a $\kappa$-sized good set of indiscernibles for the model $\langle L_\kappa[A], A\rangle$. [3]

$M$-ultrafilters: Suppose a transitive $M\models {\rm ZFC}^-$, the theory ${\rm ZFC}$ without the power set axiom (and using collection and separation rather than merely replacement) and $\kappa$ is a cardinal in $M$. We call $U\subseteq P(\kappa)^M$ an $M$-ultrafilter if the model $\langle M,U\rangle\models$“$U$ is a normal ultrafilter on $\kappa$”. In the case when the $M$-ultrafilter is not an element of $M$, the model $\langle M,U\rangle$ of $M$ together with a predicate for $U$ often fails to satisfy much of ${\rm ZFC}$. An $M$-ultrafilter $U$ is said to be weakly amenable (to $M$) if for every $A\in M$ of size $\kappa$ in $M$, the intersection $A\cap U$ is an element of $M$. An $M$-ultrafilter $U$ is countably complete if every countable sequence (possibly external to $M$) of elements of $U$ has a non-empty intersection (even if the intersection is not itself an element of $M$). A weak $\kappa$-model is a transitive set $M\models {\rm ZFC}^- $ of size $\kappa$ and containing $\kappa$ as an element. A modified ultrapower construction using only functions on $\kappa$ that are elements of $M$ can be carried out with an $M$-ultrafilter. If the $M$-ultrafilter happens to be countably complete, then the standard argument shows that the ultrapower is well-founded. If the $M$-ultrafilter is moreover weakly amenable, then a weakly amenable ultrafilter on the image of $\kappa$ in the well-founded ultrapower can be constructed from images of the pieces of $U$ that are in $M$. The ultrapower construction may be iterated in this manner, taking direct limits at limit stages, and in this case the countable completeness of the $M$-ultrafilter ensures that every stage of the iteration produces a well-founded model. [4] (Ch. 19)

A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists a weakly amenable countably complete $M$-ultrafilter on $\kappa$. [3]

Contents

Ramsey cardinals and the constructible universe

Ramsey cardinals imply that $0^\sharp$ exists and hence there cannot be Ramsey cardinals in $L$. [4]

Relations with other large cardinals

  • Measurable cardinals are Ramsey and stationary limits of Ramsey cardinals. [1]
  • Ramsey cardinals are unfoldable (using the $M$-ultrafilters characterization) and stationary limits of unfoldable cardinals (as they are stationary limits of $\omega_1$-[[$\alpha$-iterable|iterable]] cardinals).
  • Ramsey cardinals are stationary limits of completely ineffable cardinals, they are weakly ineffable but but the least Ramsey cardinal is not ineffable. [5]

Ramsey cardinals and forcing

  • Ramsey cardinals are preserved by small forcing. [4]
  • Ramsey cardinals $\kappa$ are preserved by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. [6]
  • If $\kappa$ is Ramsey, there is a forcing extension in which $\kappa$ remains Ramsey and $2^\kappa\gt\kappa$. Indeed, if the ${\rm GCH}$ holds and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and satisfying $F(\alpha)\leq F(\beta)$ for $\alpha<\beta$ and $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey and $2^\delta=F(\delta)$ for every regular cardinal $\delta$. [7]
  • If the existence of Ramsey cardinals is consistent with ZFC, then there is a model of ZFC in which $\kappa$ is not Ramsey, but becomes Ramsey in a forcing extension. [6]

Strongly Ramsey cardinal

Strongly Ramsey cardinals were introduced by Gitman in [5]. They strengthen the $M$-ultrafilters characterization of Ramsey cardinals from weak $\kappa$-models to $\kappa$-models. A weak $\kappa$-model $M$ is a $\kappa$-model if additionally $M^{\lt\kappa}\subseteq M$. A cardinal $\kappa$ is strongly Ramsey if every $A\subseteq\kappa$ is contained in a $\kappa$-model $M$ for which there exists a weakly amenable $M$-ultrafilter on $\kappa$. An $M$-ultrafilter for a $\kappa$-model $M$ is automatically countably complete since $\langle M,U\rangle$ satisfies that it is $\kappa$-complete and it must be correct about this since $M$ is closed under sequences of length less than $\kappa$.

  • Measurable cardinas are strongly Ramsey.
  • Strongly Ramsey cardinals are Ramsey and stationary limits of Ramsey cardinals.
  • The least Ramsey cardinal is not ineffable. [5]
  • Forcing related properties of strongly Ramsey cardinals are the same as those of Ramsey cardinals described above. [6]

Virtually Ramsey cardinal

Virtually Ramsey cardinals were introduced by Sharpe and Welch in [8]. They weaken the good indiscernibles characterization of Ramsey cardinals and were motivated by finding an upper bound on the consistency strength of a variant of Chang's Conjecture studied in [8]. For $A\subseteq\kappa$, define that $\mathscr I=\{\alpha<\kappa\mid$ there is an unbounded good set of indiscernibles $I_\alpha\subseteq\alpha$ for $\langle L_\kappa[A],A\rangle\}$. A cardinal $\kappa$ is virtually Ramsey if for every $A\subseteq\kappa$, the set $\mathscr I$ contains a club of $\kappa$.

Virtually Ramsey cardinals are Mahlo and a virtually Ramsey cardinal that is weakly compact is already Ramsey. It is consistent from a Ramsey cardinal that there is a virtually Ramsey cardinal that is not Ramsey. It is open whether virtually Ramsey cardinals are weaker than Ramsey cardinals. [9]

$\alpha$-iterable cardinal

The $\alpha$-iterable cardinals for $1\leq\alpha\leq\omega_1$ were introduced by Gitman in [9]. They form a hierarchy of large cardinal notions strengthening weakly compact cardinals, while weakening the $M$-ultrafilters characterization of Ramsey cardinals. Recall that if $\kappa$ is Ramsey, then every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $M$-ultrafilter, the ultrapower construction with which may be iterated through all the ordinals. Suppose $M$ is a weak $\kappa$-model and $U$ is an $M$-ultrafilter on $\kappa$. Define that:

  • $U$ is $0$-good if the ultrapower is well-founded,
  • $U$ is 1-good if it is 0-good and weakly amenable,
  • for an ordinal $\alpha>1$, $U$ is $\alpha$-good, if it produces at least $\alpha$-many well-founded iterated ultrapowers.

Using a theorem of Gaifman [10], if an $M$-ultrafilter is $\omega_1$-good, then it is already $\alpha$-good for every ordinal $\alpha$.

For $1\leq\alpha\leq\omega_1$, a cardinal $\kappa$ is $\alpha$-iterable if every $A\subseteq\kappa$ is contained in a weak $\kappa$-model $M$ for which there exists an $\alpha$-good $M$-ultrafilter on $\kappa$. The $\alpha$-iterable cardinals form a hierarchy of strength above weakly compact cardinals and below Ramsey cardinals.

  • $1$-iterable cardinals are weakly ineffable and stationary limits of completely ineffable cardinals. The least $1$-iterable cardinal is not ineffable. [5]
  • An $\alpha$-iterable cardinal is $\beta$-iterable and a stationary limit of $\beta$-iterable cardinals for every $\beta<\alpha$. [9]
  • A Ramsey cardinal is $\omega_1$-iterable and a stationary limit of $\omega_1$-iterable cardinals. This is already true of an $\omega_1$- Erdős cardinal. [8]
  • It is consistent from an $\omega$- Erdős cardinal that for every $n\in\omega$, there is a proper class of $n$-iterable cardinals.
  • A $2$-iterable cardinal is a limit of remarkable cardinals. [9]
  • A remarkable cardinal implies the consistency of a $1$-iterable cardinal. [9]
  • $\omega_1$-iterable cardinals imply that $0^\sharp$ exists and hence there cannot be $\omega_1$-iterable cardinals in $L$. For $L$-countable $\alpha$, the $\alpha$-iterable cardinals are downward absolute to $L$. In fact, if $0^\sharp$ exists, then every Silver indiscernible is $\alpha$-iterable in $L$ for every $L$-countable $\alpha$. [9]
  • $\alpha$-iterable cardinals $\kappa$ are preserved by small forcing, by the canonical forcing of the ${\rm GCH}$, by fast function forcing, and by the forcing to add a slim $\kappa$-Kurepa tree. If $\kappa$ is $\alpha$-iterable, there is a forcing extension in which $\kappa$ remains $\alpha$-iterable and $2^\kappa\gt\kappa$. [11]

References

  1. Erdős, Paul and Hajnal, Andras. Some remarks concerning our paper ``On the structure of set-mappings''. Non-existence of a two-valued $\sigma $-measure for the first uncountable inaccessible cardinal. Acta Math Acad Sci Hungar 13:223--226, 1962. MR   bibtex
  2. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) bibtex

  3. Dodd, Anthony and Jensen, Ronald. The core model. Ann Math Logic 20(1):43--75, 1981. www   DOI   MR   bibtex

  4. Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www   bibtex

  5. Gitman, Victoria. Ramsey-like cardinals. The Journal of Symbolic Logic 76(2):519-540, 2011. www   arχiv   MR   bibtex

  6. Gitman, Victoria and Johnstone, Thomas. Indestructibility for Ramsey and Ramsey-like cardinals. (In preparation) bibtex

  7. Cody, Brent and Gitman, Victoria. Easton's theorem for Ramsey and strongly Ramsey cardinals. (In preparation) bibtex

  8. Sharpe, Ian and Welch, Philip. Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann Pure Appl Logic 162(11):863--902, 2011. www   DOI   MR   bibtex

  9. Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www   arχiv   MR   bibtex

  10. Gaifman, Haim. Elementary embeddings of models of set-theory and certain subtheories. Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), pp. 33--101, Providence R.I., 1974. MR   bibtex

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